The first fundamental form to compute the lenght of a curve and the angle between tw0 curves
You've set this up well, except for one issue. All you have to do is realize where the vertices of the triangle are (in $(u,v)$ coordinates). One is at $(0,0)$, one is at $(0,a)$, and the last is at $(\sinh a,a)$. So are your curves the correct boundary curves of the triangle? (Hint: One is wrong.) But once you know where the vertices are, there's no problem computing the first fundamental form at this points to find angles.